   Chapter 3.8, Problem 2E

Chapter
Section
Textbook Problem

A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician Theodor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 50 cells.(a) Find the 1elative growth rate.(b) Find an expression for the number of cells after t hours.(c) Find the number of cells after 6 hours.(d) Find the rate of growth after 6 hours.(c) When will the population reach a million cells?

(a)

To determine

To find: The relative growth rate.

Explanation

Given:

The initial population of a culture is p0=50.

Theorem used:

“The only solutions of the differential equation dydt=ky are the exponential functions

y(t)=y(0)ekt.”

Calculation:

Let P(t) be the population of the bacterium Escherichia coli.

Given that,

The population doubles every 20 minutes, P(13)=2p0.

P(13)=2p0=2(50)       (Qp0=50)=100

Obtain the relative growth rate

(b)

To determine

To find: The number of cells after t hours.

(c)

To determine

To find: The number of cells after 6 hours. That is,P(6).

(d)

To determine

To find: The rate of growth after 6 hours. That is,P(6).

(e)

To determine

To find: The time at which population reach a million cells.

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